Introduction:

 

In the last few years there have been significant advances in the characterization of the mechanical properties of carbon nanotubes, silicon and silicon oxide surfaces and cantilevers [1-4].  Properties such as resonance frequency and bending modulus have been extensively researched for both cantilevers and carbon nanotubes [1, 5-8].   A great deal of work has also been done on the mechanical properties of functionalized cantilevers, generally covered on one face with a metallic layer of variable thickness and composition, polymer coatings, or monolayers of self-assembled or chemically bonded molecules of various types [8-16].  Functionalized cantilevers exhibit deflection due to differences in surface stress between functionalized and non-functionalized opposite faces.  The surface stress on the functionalized face is a function of the environment, and the changes induced by the environment are generally reversible.  Since it is possible to control the environment, it is also possible to generate controlled deflection of the cantilever. This presents significant opportunities to utilize functionalized cantilevers as reliable and reversible actuators in nanomechanical devices such as valves, pumps, switches, etc.

 

In this paper we present the design of a fluid control valve that utilizes a silicon cantilever, functionalized with a covalently bonded monolayer of acrylic acid, as the actuator that opens and closes flow through a fluid conduit, a single-wall carbon nanotube (SWCNT).  The on/off position of the valve is controlled by pH changes in the surrounding environment.  Changes in pH affect the charge of the organic acid groups bonded to the surface of the cantilever.  The electrostatic energy of these acid groups on the functionalized surface of the cantilever causes a compressive stress that deflects it to the closed position [10].  The device assembly and valve components are feasible with today's laboratory synthesis capabilities (SWCNT synthesis methods, silicon etching techniques and covalent monolayer assembly). 

 

Classical engineering design approximations can be utilized to lower the computational costs of current molecular modeling methodologies. As devices become larger, their design becomes computationally more expensive and in some cases impractical (an N2 problem).  In the present case, for example, the system has in excess of 75,000 atoms and is over 30 nm long.  Since the performance of the system depends on the electrostatic interaction of all the functional molecules on the surface of the device, it is necessary to include in the calculations all the electrostatic interactions of all charged particles in order to have an accurate model.  Efficient electrostatic lattice sum methods, such as Ewald and Particle-Mesh Ewald, cannot be employed without introducing artifacts due to imposition of periodic boundary conditions (the device under consideration is not a periodic system), thus we are left with a direct electrostatic sum in real space.  In most standard molecular simulation packages this would require including all non-bonded interactions within a radius of 35 nm. Since the number of non-bonded interactions scales as N2, this makes the calculations unnecessarily lengthy assuming that the computing system has enough memory to store such a large energy expression.   Furthermore, the use of cutoffs or splines in the calculation of the electrostatic energy can underestimate the correct values by factors of one order of magnitude for the monolayer dimensions considered here.

 

A second computational issue is the statistical nature of the evaluation of the convergence criteria used in a molecular mechanics simulation.  Average forces or strain energies don't necessarily represent the equilibrium state of the system, local or global.  Due to the large number of atoms, large residual forces and stresses may be present in a small section of the system, while the average force and strain are quite small.  A criteria based on average forces/strains may leave the device in a non-equilibrium state. 

 

These difficulties will be overcome with more powerful computer systems and more flexible molecular simulations software packages, but this requires time, human and financial resources, and breakthroughs in computer systems.  A more practical approach is to perform a semi-continuum characterization of the system, whereby each component of a particular device is individually characterized using molecular simulations prior to a classical analysis of the entire device.  This characterization includes the determination of the classical engineering parameters needed for the continuum analysis, such as natural vibrational frequencies, elasticity moduli, points of mechanical failure, etc., at the length scale under consideration.  Once these parameters are available, the continuum analysis of the assembled device becomes a simple exercise.  The latter approach is presented in this paper.

 

It must also be noted that in order to have a complete ab initio description of the device presented here, there are several fundamental issues that still need to be answered.  Two of them are the molecular transport phenomena through SWCNT's and the solvation properties of acid monolayers on deflected cantilevers.  Although no attempt is made to resolve these issues in this paper, the engineering method presented here remains a useful design tool for the future inclusion of these effects. This is because it is generally possible to find smooth-varying mathematical approximations of those effects, over finite length and time scales, which can then be incorporated into the engineering continuum models.  Atomistic simulations provide the range over which these smooth approximations are valid as well as identify the regions where transitions take place.  A prime example of a complex system which can be classically approximated with segment-wise smoothly varying functions is the bending of a SWCNT, which exhibits buckling phenomena (see figure 9).

 

 


 


Figure 9:  17,17 carbon nanotube at a curvature of 0.0031/Ang (beyond the point of buckling).  

 



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